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Ineluctable modality of the
distributed
On Joseph Halpern’s work on
knowledge in distributed
systems
Peter Alvaro
UC Berkeley
choose-your-own-adventure talk
Last time at PWL…
•  The agreement problem(s)
•  Impossibility results
•  A “weakest” failure detector
Today: knowledge	
  
It’s not just for byzantine stuff
I'm not a great fool, so I can clearly not choose the wine in
front of you. But you must have known I was not a great
fool; you would have counted on it, so I can clearly not
choose the wine in front of me.
Why you should care
A correct distributed program achieves
(nontrivial) distributed property X.
Some tricky questions before we start coding:
1.  Is X even attainable?
2.  Cheapest protocol that gets me X?
3.  How should I implement it?
A strong claim about distributed
correctness properties	
  
Uncertainty is what makes reasoning about
distributed systems difficult.
Uncertainty is the abundance of possibilities.
Knowledge is the dual of possibility
A strong statement about
protocols
How: Protocols just describe what actions to
take based on local knowledge.
Why: Protocols are just mechanisms to
ensure that a group has shared knowledge of
a fact.
A good paper about bridging the gap
between properties and protocols
For example
•  Commit protocols
– each agent knows the commit/abort
decision AND knows that all agents know
the decision
•  Distributed garbage collection
– an agent knows that no remote references
exist to a particular object, and that all other
agents know
For example
•  When the leader has received phase 2b messages for
value v and ballot bal from a majority of the acceptors, it
knows that the value v has been chosen. [paxos]
•  a process takes a checkpoint when it knows that all
processes on which it computationally depends took their
checkpoints [An Efficient Protocol for Checkpointing
Recovery in Distributed Systems, Kim and Park]
•  and therefore a cohort with a later viewstamp for some
view knows everything known to a cohort with an earlier
viewstamp for that view. [viewstamped replication]
•  Since each member of Si serves as an arbitrator, the
requesting node knows that it is the only node that has
been granted mutual exclusion [A sqrt(N) Algorithm for
Mutual Exclusion in Decentralized Systems, Maekawa]
Warmup: RPC protocols
Hi!
Alice Bob
Warmup: RPC protocols
Hi!
Alice Bob
Issue: uncertainty!
Uncertain environment è Uncertain outcomes
Warmup: RPC protocols
Alice Bob
Issue: uncertainty!
Uncertain environment è Uncertain outcomes
Warmup: RPC protocols
Hi!
Retry	
  
Alice Bob
Warmup: RPC protocols
Hi!
Retry	
  
Alice Bob
Warmup: RPC protocols
Hi!
Retry	
  
Alice Bob
Warmup: RPC protocols
Hi!
Retry	
  
Alice Bob
Warmup: RPC protocols
Hi!
Retry	
  
Alice Bob
Warmup: RPC protocols
Hi!
Issues: infinite (sender) behavior & state,
at-least-once delivery
Retry	
  
Alice Bob
Warmup: RPC protocols
Hi!
Retry with ACKS
Hi!
Alice Bob
Warmup: RPC protocols
Hi!
Retry with ACKS
Hi!Hi!
Alice Bob
Hi!
Warmup: RPC protocols
Hi!
Hi yourself
Retry with ACKS
Hi!
Issues: at-least once delivery
Hi!
Alice Bob
Hi!
Warmup: RPC protocols
Hi!
Hi yourself
Retry with ACKS
Hi!
Issues: at-least once delivery
Hi!
Alice Bob
Warmup: RPC protocols
Retry with ACKS
Issues: at-least once delivery
Alice Bob
Hi!
a	
  good	
  paper	
  about	
  principled	
  distributed	
  GC	
  
Warmup: RPC protocols
Hi!
Issues: infinite receiver state
Receiver buffers, dedups
Alice Bob
Warmup: RPC protocols
Issues: infinite receiver state
Hi!
Receiver buffers, dedups
Alice Bob
Warmup: RPC protocols
Hi!
ACK-ACKing
Hi!
Alice Bob
Warmup: RPC protocols
Hi!
Hi yourself
ACK-ACKing
Hi!
Issue: uncertainty
Alice Bob
Warmup: RPC protocols
Hi!
Hi yourself
ACK-ACKing
Hi!
Alice Bob
Warmup: RPC protocols
ACK-ACKing
Hi!
Alice Bob
Ahoy	
  
Warmup: RPC protocols
ACK-ACKing
Hi!
Alice Bob
Ahoy	
  
Warmup: RPC protocols
ACK-ACKing
Alice Bob
Warmup: RPC protocols
ACK-ACKing
Issue: uncertainty
Alice Bob
Warmup: RPC protocols
Issues: infinite hot potato
Alice Bob
Warmup: RPC protocols
Issues: infinite hot potato
Alice Bob
Warmup: RPC protocols
Issues: infinite hot potato
Alice Bob
Warmup: RPC protocols
Issues: infinite hot potato
Alice Bob
what does this remind me of?
Refresher: the two generals problem
Logic time
(propositional) logic
ϕ ϕ if ϕ is atomic
ϕ ∧ ψ true if both ϕ and ψ are true
¬ϕ true if ϕ is false
Sweet duality:
ϕ ∨ ψ = ¬(¬ϕ ∧ ¬ψ)
ϕ ⇒ ψ= ¬(ϕ ∧ ¬ψ)
q ⇒ p
p = “the write is stable”
q = “the write is acknowledged”
modality, duality
∃xϕ === ¬∀x ¬ϕ
¯ϕ === ¬£¬ϕ
Symbol	
   Temporal	
   Deon/c	
   Epistemic	
  
¯	
   Some8mes	
   Is	
  permi:ed	
   Is	
  possible	
  
£	
   Always	
   Is	
  obligatory	
   Is	
  known	
  
Knowledge is the dual of possibility
Epistemic modal logic
ϕ = “the write is stable”
Kaliceϕ = “alice knows ϕ”
KaliceKbobϕ = “alice knows bob knows ϕ”
KaliceKbobKcarolϕ = “alice knows bob knows
carol knows ϕ”
[…]
Epistemic modal logic
ϕ = “the write is stable”
Eϕ = “everyone* knows ϕ”
EEϕ = “everyone knows everyone knows ϕ”
[…]
A driver will not feel safe going when he sees a
green light unless he knows that everyone else
knows and follows the rules.
Common knowledge
ϕ = “the write is stable”
Eϕ = “everyone* knows ϕ”
EEϕ = “everyone knows everyone knows ϕ”
[…]
Eiϕ = “(everyone knows * i) ϕ”
Cϕ = E∞ϕ = “it is common knowledge that ϕ”
Distributed knowledge
ϕ = “the write is stable”
Dϕ = “ϕ is implicitly known by the group”
Sϕ = “someone knows ϕ”
Protocols	
  climb	
  the	
  hierarchy	
  
Cϕ
[…]
Ek+1ϕ
[…]
Eϕ
Sϕ
Dϕ
ϕ	
  
Protocols	
  climb	
  the	
  hierarchy	
  
Cϕ
[…]
Ek+1ϕ
[…]
Eϕ
Sϕ
Dϕ
ϕ	
  
Deadlock detection
ϕ is distributed knowledge	
  
Someone knows ϕ
Protocols	
  climb	
  the	
  hierarchy	
  
Cϕ
[…]
Ek+1ϕ
[…]
Eϕ
Sϕ
Dϕ
ϕ	
  
Reliable broadcast
Someone knows ϕ
ϕ is distributed knowledge	
  
Everyone knows ϕ
Protocols	
  climb	
  the	
  hierarchy	
  
Cϕ
[…]
E3ϕ
E2ϕ
Eϕ
Sϕ
Dϕ
ϕ	
  
Uniform
Reliable broadcast
Someone knows ϕ
ϕ is distributed knowledge	
  
Everyone knows ϕ
Everyone knows
everyone knows ϕ
Protocols	
  climb	
  the	
  hierarchy	
  
Cϕ
[…]
E3ϕ
E2ϕ
Eϕ
Sϕ
Dϕ
ϕ	
  
Someone knows ϕ
ϕ is distributed knowledge	
  
Everyone knows ϕ
Everyone knows
everyone knows ϕ
Some crazy BFT
protocol
(Everyone knows)k ϕ
Protocols	
  climb	
  the	
  hierarchy	
  
Cϕ
[…]
E3ϕ
E2ϕ
Eϕ
Sϕ
Dϕ
ϕ	
  
Knowledge	
  Highway	
  
E10ϕ	
   	
   	
   	
   	
  10	
  
E100ϕ 	
   	
   	
  	
  	
  	
  	
  	
  	
  	
  100	
  
Cϕ 	
  ∞	
  
Applications of knowledge
A correct distributed program achieves
(nontrivial) distributed property X.
Some tricky questions before we start coding:
1.  Is X even attainable?
2.  Cheapest protocol that gets me X?
3.  How should I implement it?
Applications: impossibility
“in a system in which communication is not
guaranteed, common knowledge of
initially-undetermined facts is not
attainable in any run of any protocol.”
Corollary: the 2 generals problem is
unsolvable
Let’s use knowledge to prove it!
But first… lots of formalism to get through L
Road map for the proof:
1.  Semantics of modal logic
2.  Distributed system model
3.  A quick and easy lemma
4.  Big theorem: Common knowledge is not
attainable via protocol
5.  Lemma 2: if the generals attack, they have
common knowledge of the attack.
6.  Corollary: 2 generals is unsolvable
Semantics
Semantics: structures
Formulae are well-formed, meaningless
strings of symbols
Structures give meaning to formulae
(in the very narrow sense of making them all either true or false)
S |= ϕ
Semantics:
propositional structures
Propositional formula:
S |= p ∧ q
Need:
1.  a map S from variable names to T/F
2.  rules; e.g. S |= ϕ ∧ ψ iff S |= ϕ and S |= ψ
Semantics:
first-order structures
First-order formula:
S |= ∀x, dog(x) ⇒ big(x) ∧ likes(x, me)
Need:
1.  S assigns “records” to dog, big and likes.
2.  Rules; e.g. S |= ∀xφ iff for all d ∈	
  |S|,	
  S[x	
  :=	
  d]	
  |=	
  φ	
  
Semantics:
first-order structures
•  First-order logic:
S |= ∀x, dog(x) ⇒ big(x) ∧ likes(x, me)
dog	
  
Rex	
  
Fido	
  
Rover	
  
big	
  
Rex	
  
Fido	
  
me	
  
likes	
  
Rex	
   me	
  
Fido	
   me	
  
Rover	
   me	
  
me	
   me	
  
couple good papers about using FO
logic to program distributed systems
Semantics – modal logic
S |= (£¬p) ∧ (q ⇒ ¯r)
Need: a structure that can interpret the
propositional formulae under different modalities
Kripke structure: (W, π, R)
•  W is a set of worlds
•  For each element of W, π is a propositional structure
•  R is an accessibility relation among elements of W
S1	
   S3	
  
Semantics – modal logic
Temporal logic
S |= (£¬p) ∧ (q ⇒ ¯r)
	
  q	
  	
  	
  r	
  
r	
  q	
  
S1	
   S3	
  
S2	
  
Kripke structure: (W, π, R) 	
  
Semantics – modal logic
Epistemic logic
S |= r ∧ ¬Kir ∧ Ki(Kjr or Kj¬r) ∧ Kjr ∧ ¬Kj¬Kir
	
  q	
  	
  	
  r	
  
r	
  q	
  
S1	
   S3	
  
S2	
  
i	
   j	
  
Kripke structure: (W, π, Ri) 	
  
a model of distributed systems
(r,t)
p1 p2 p3 p4 Idealized time
}h(p4,r,t)
A run
r ∈ R
Knowledge-based interpretations
Knowledge interpretation: I = (R, π, {v1,v2,[..]})
Knowledge point: (I, r, t)
R – a set of runs
π – assigns a truth assignment to propositions
for each point in R
vi – A view function for R for some agent i
(determined by h)
Kripke structure: (W, π, R) 	
  
Truth in a knowledge interpretation
(I,r,t) |= φ iff π(r,t)(φ) = true
(If φ is a ground formula)
(I,r,t) |= ¬φ iff (I,r,t) |= φ
(I,r,t) |= φ ∧ ψ iff (I,r,t) |= φ and (I,r,t) |= ψ
(I,r,t) |= Kiφ iff (I,r’,t’) |= φ for all (r’,t’) in R
satisfying v(pi,r,t) = v(pi,r’,t’)	
  
(I,r,t) |= Eφ iff (I,r’,t’) |= Kiφ for all pi
(I,r,t) |= Cφ iff (I,r’,t’) |= Ekφ for all k
choose-your-own-adventure
•  If you’d like to gloss over
the proof and skip to other
applications of knowledge,
turn to page 62
•  If you’d like to dive into the
weeds, turn to page 54.
Truth in a knowledge interpretation
(I,r,t) |= Cφ iff (I,r’,t’) |= Ekφ for all k
Fixed point axiom:
Cφ = E(φ ∧ Cφ)
Induction rule:
From φ ⇒ E(φ ∧ ψ) infer φ ⇒ Cψ
communication is not guaranteed	
  
NG1: For all runs r and times t, there exists a
run r’ extending (r,t) such that […] no messages
are received in r’ at or after time t.
NG2: If in run r processor pi does not receive
any messages in the interval (t’,t), then there is
a run r’ extending (r,t’) such that […] h(pi,r,t’’) =
h(pi,r’,t’’) for all t’’ < t, and no processor pj != pi
receives a message in r’ in the interval (t’,t).
	
  
Lemma 1
If, in two different runs (r and r’) of the same
protocol, some h(p, r, t) = h(p, r’, t), then
(I, r, t) |= Cφ iff (I, r’, t) |= Cφ
Sorry, no proof today!
Common knowledge is not attainable in a system in
which communication is not guaranteed
Take runs r and r- in R, with the same initial
configuration, s.t. no messages are received in r-
up till time t. Then (I,r,t) |= Cφ iff (I,r-,t) |= Cφ.
Proof (by induction on d(r)*): 	
  
•  Base case: d(r)=0. h(p1,r,t) = h(p1,r-,t). By Lemma
1, (I,r,t) |= Cφ iff (I,r-,t) |= Cφ.
*	
  d(r)	
  is	
  the	
  number	
  of	
  messages	
  received	
  in	
  run	
  r.	
  
Common knowledge is not attainable in a system in
which communication is not guaranteed
Inductive case: d(r) = k+1. Let:	
  
•  t’ < t -- the latest time a message is received in r before t.
•  pj -- a processor that received a message at t’
•  pi –a processor (!= pj)
By NG2, there is a run r’ extending (r,t’) s.t. h(pi,r,t’’)=h(pi,r’,t’’) for
all t’’ <= t, and all processors (besides pi) receive no messages
in the interval (t’, t).
By construction, d(r’) <= k, so by the IH (I,r’,t) |= Cφ iff (I,r-,t) |= Cφ.
But since h(pi,r,t) = h(pi,r’,t), by Lemma 1 (I,r’,t) |= Cφ iff (I,r,t) |= Cφ.
So (I,r,t) |= Cφ iff (I,r-,t) |= Cφ.
QED
Common knowledge is not attainable in a system in
which communication is not guaranteed
Review: we showed that common knowledge cannot be
gained (or lost) by exchanging messages.
Corollary: the 2 generals will never attack.
But we still need to prove one more lemma:
Any correct protocol for coordinated attack has the
property that whenever the generals attack, it is common
knowledge that they are attacking.
Lemma 2: coordinated attack
requires common knowledge
Let ψ = the generals are attacking
Assume the generals (A and B) attack at (r*, t*) – we show that
(I,r*,t*) |= Cψ.
Pick an arbitrary point (r,t). We show ψ ⇒ Eψ is valid in R.
•  If (I,r,t) |= ψ, then the generals attack at (r,t). Consider (r’,t’), in
which A has the same history at (r,t). Since the protocol is
deterministic (assumption), A must also attack in (r’,t’); since
the protocol is correct, B does also, and so (I,r’,t’) |= ψ. It
follows that (I,r,t) |= Eψ, so ψ ⇒ Eψ is valid in R.
•  If (I,r,t) |= ¬ψ, then trivially ψ ⇒ Eψ is valid in R.
By the induction rule, ψ ⇒ Cψ is valid in R
Coup de grace
ψ = the generals are attacking
1.  By assumption, Cψ does not hold if no
messages are exchanged.
2.  By theorem 1, Cψ will never hold.
3.  By lemma 2, the generals cannot attack
unless Cψ.
	
  
Phew. but…?
Common knowledge is a prerequisite for agreement.
Common knowledge is not attainable via protocol.
Halpern:
These results may seem paradoxical.
Reality check
Fragile assumptions on which the proofs rest:
•  Deterministic protocol
•  Simultaneous agreement is necessary
•  “Communication not guaranteed”
•  Lack of useful a priori common knowledge
Bootstrapping common knowledge
•  The ``weakest failure detector’’
•  Spanner’s global clock
•  Sequence wraparound
Applications of knowledge
A correct distributed program achieves
(nontrivial) distributed property X.
Some tricky questions before we start coding:
1.  Is X even attainable?
2.  Cheapest protocol that gets me X?
3.  How should I implement it?
lower bounds for protocols
[Hadzilacos, PODS’87]: A knowledge-theoretic
analysis of atomic commitment protocols
1.  All of the variants of 2pc ((de-)centralized,
linear/nested, etc) are identical from a
knowledge perspective
2.  All 2PC variants attain the minimum level of
knowledge needed to commit
3.  3PC attains the minimum needed to commit
without blocking
4.  Lower bound for messages: nested 2PC.
A good paper about automatically
choosing cheap coordination mechanisms
Applications of knowledge
A correct distributed program achieves
(nontrivial) distributed property X.
Some tricky questions before we start coding:
1.  Is X even attainable?
2.  Cheapest protocol that gets me X?
3.  How should I implement it?
protocol implementation / synthesis
•  Halpern and Fagin: knowledge-based programming
[PODC’95]
	
  
case	
  of	
  	
  
	
  K(Msg)	
  and	
  (KE(AckedMsg))	
  do	
  deliver(Msg)	
  
	
  K(Msg)	
  and	
  !KE(AckedMsg)	
  do	
  relay(Msg)	
  	
  	
  
end	
  
•  Matteo interlandi [Datalog2.0’11]:
Knowlog: knowledge-enriched Dedalus
	
  log(Tx_id,"abort")@next	
  :-­‐	
  Dvote(Vote,Tx_id),Vote=="no",	
   	
   	
  	
  	
  
	
  	
  	
  	
  	
  par8cipants(X),transac8on(Tx_id,State),State=="vote-­‐req".	
  	
  
A good paper about Dedalus
Monotonicity and knowledge
Monotonic:
the more you know,
the more you know.
Cϕ
[…]
E3ϕ
E2ϕ
Eϕ
Sϕ
Dϕ
ϕ	
  
A good paper about monotonicity
and distributed consistency
Remember
•  Knowledge is the dual of possibility
•  Local knowledge dictates protocol
behavior
•  The purpose of protocols is obtaining a
particular level of distributed knowledge
•  Deep connections between semantic
structures and system behavior
•  Common knowledge is unattainable via
protocol (but there is still hope)
Protocols	
  climb	
  the	
  hierarchy	
  
Cϕ
[…]
E3ϕ
E2ϕ
Eϕ
Sϕ
Dϕ
ϕ	
  
Knowledge	
  Highway	
  
E10ϕ	
   	
   	
   	
   	
  10	
  
E100ϕ 	
   	
   	
  	
  	
  	
  	
  	
  	
  	
  100	
  
Cϕ 	
  ∞	
  

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Pwl rewal-slideshare

  • 1. Ineluctable modality of the distributed On Joseph Halpern’s work on knowledge in distributed systems Peter Alvaro UC Berkeley
  • 3. Last time at PWL… •  The agreement problem(s) •  Impossibility results •  A “weakest” failure detector Today: knowledge  
  • 4. It’s not just for byzantine stuff I'm not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool; you would have counted on it, so I can clearly not choose the wine in front of me.
  • 5. Why you should care A correct distributed program achieves (nontrivial) distributed property X. Some tricky questions before we start coding: 1.  Is X even attainable? 2.  Cheapest protocol that gets me X? 3.  How should I implement it?
  • 6. A strong claim about distributed correctness properties   Uncertainty is what makes reasoning about distributed systems difficult. Uncertainty is the abundance of possibilities. Knowledge is the dual of possibility
  • 7. A strong statement about protocols How: Protocols just describe what actions to take based on local knowledge. Why: Protocols are just mechanisms to ensure that a group has shared knowledge of a fact.
  • 8. A good paper about bridging the gap between properties and protocols
  • 9. For example •  Commit protocols – each agent knows the commit/abort decision AND knows that all agents know the decision •  Distributed garbage collection – an agent knows that no remote references exist to a particular object, and that all other agents know
  • 10. For example •  When the leader has received phase 2b messages for value v and ballot bal from a majority of the acceptors, it knows that the value v has been chosen. [paxos] •  a process takes a checkpoint when it knows that all processes on which it computationally depends took their checkpoints [An Efficient Protocol for Checkpointing Recovery in Distributed Systems, Kim and Park] •  and therefore a cohort with a later viewstamp for some view knows everything known to a cohort with an earlier viewstamp for that view. [viewstamped replication] •  Since each member of Si serves as an arbitrator, the requesting node knows that it is the only node that has been granted mutual exclusion [A sqrt(N) Algorithm for Mutual Exclusion in Decentralized Systems, Maekawa]
  • 12. Warmup: RPC protocols Hi! Alice Bob Issue: uncertainty! Uncertain environment è Uncertain outcomes
  • 13. Warmup: RPC protocols Alice Bob Issue: uncertainty! Uncertain environment è Uncertain outcomes
  • 19. Warmup: RPC protocols Hi! Issues: infinite (sender) behavior & state, at-least-once delivery Retry   Alice Bob
  • 20. Warmup: RPC protocols Hi! Retry with ACKS Hi! Alice Bob
  • 21. Warmup: RPC protocols Hi! Retry with ACKS Hi!Hi! Alice Bob Hi!
  • 22. Warmup: RPC protocols Hi! Hi yourself Retry with ACKS Hi! Issues: at-least once delivery Hi! Alice Bob Hi!
  • 23. Warmup: RPC protocols Hi! Hi yourself Retry with ACKS Hi! Issues: at-least once delivery Hi! Alice Bob
  • 24. Warmup: RPC protocols Retry with ACKS Issues: at-least once delivery Alice Bob Hi!
  • 25. a  good  paper  about  principled  distributed  GC  
  • 26. Warmup: RPC protocols Hi! Issues: infinite receiver state Receiver buffers, dedups Alice Bob
  • 27. Warmup: RPC protocols Issues: infinite receiver state Hi! Receiver buffers, dedups Alice Bob
  • 29. Warmup: RPC protocols Hi! Hi yourself ACK-ACKing Hi! Issue: uncertainty Alice Bob
  • 30. Warmup: RPC protocols Hi! Hi yourself ACK-ACKing Hi! Alice Bob
  • 35. Warmup: RPC protocols Issues: infinite hot potato Alice Bob
  • 36. Warmup: RPC protocols Issues: infinite hot potato Alice Bob
  • 37. Warmup: RPC protocols Issues: infinite hot potato Alice Bob
  • 38. Warmup: RPC protocols Issues: infinite hot potato Alice Bob
  • 39. what does this remind me of? Refresher: the two generals problem
  • 41. (propositional) logic ϕ ϕ if ϕ is atomic ϕ ∧ ψ true if both ϕ and ψ are true ¬ϕ true if ϕ is false Sweet duality: ϕ ∨ ψ = ¬(¬ϕ ∧ ¬ψ) ϕ ⇒ ψ= ¬(ϕ ∧ ¬ψ) q ⇒ p p = “the write is stable” q = “the write is acknowledged”
  • 42. modality, duality ∃xϕ === ¬∀x ¬ϕ ¯ϕ === ¬£¬ϕ Symbol   Temporal   Deon/c   Epistemic   ¯   Some8mes   Is  permi:ed   Is  possible   £   Always   Is  obligatory   Is  known   Knowledge is the dual of possibility
  • 43. Epistemic modal logic ϕ = “the write is stable” Kaliceϕ = “alice knows ϕ” KaliceKbobϕ = “alice knows bob knows ϕ” KaliceKbobKcarolϕ = “alice knows bob knows carol knows ϕ” […]
  • 44. Epistemic modal logic ϕ = “the write is stable” Eϕ = “everyone* knows ϕ” EEϕ = “everyone knows everyone knows ϕ” […] A driver will not feel safe going when he sees a green light unless he knows that everyone else knows and follows the rules.
  • 45. Common knowledge ϕ = “the write is stable” Eϕ = “everyone* knows ϕ” EEϕ = “everyone knows everyone knows ϕ” […] Eiϕ = “(everyone knows * i) ϕ” Cϕ = E∞ϕ = “it is common knowledge that ϕ”
  • 46. Distributed knowledge ϕ = “the write is stable” Dϕ = “ϕ is implicitly known by the group” Sϕ = “someone knows ϕ”
  • 47. Protocols  climb  the  hierarchy   Cϕ […] Ek+1ϕ […] Eϕ Sϕ Dϕ ϕ  
  • 48. Protocols  climb  the  hierarchy   Cϕ […] Ek+1ϕ […] Eϕ Sϕ Dϕ ϕ   Deadlock detection ϕ is distributed knowledge   Someone knows ϕ
  • 49. Protocols  climb  the  hierarchy   Cϕ […] Ek+1ϕ […] Eϕ Sϕ Dϕ ϕ   Reliable broadcast Someone knows ϕ ϕ is distributed knowledge   Everyone knows ϕ
  • 50. Protocols  climb  the  hierarchy   Cϕ […] E3ϕ E2ϕ Eϕ Sϕ Dϕ ϕ   Uniform Reliable broadcast Someone knows ϕ ϕ is distributed knowledge   Everyone knows ϕ Everyone knows everyone knows ϕ
  • 51. Protocols  climb  the  hierarchy   Cϕ […] E3ϕ E2ϕ Eϕ Sϕ Dϕ ϕ   Someone knows ϕ ϕ is distributed knowledge   Everyone knows ϕ Everyone knows everyone knows ϕ Some crazy BFT protocol (Everyone knows)k ϕ
  • 52. Protocols  climb  the  hierarchy   Cϕ […] E3ϕ E2ϕ Eϕ Sϕ Dϕ ϕ   Knowledge  Highway   E10ϕ          10   E100ϕ                    100   Cϕ  ∞  
  • 53. Applications of knowledge A correct distributed program achieves (nontrivial) distributed property X. Some tricky questions before we start coding: 1.  Is X even attainable? 2.  Cheapest protocol that gets me X? 3.  How should I implement it?
  • 54. Applications: impossibility “in a system in which communication is not guaranteed, common knowledge of initially-undetermined facts is not attainable in any run of any protocol.” Corollary: the 2 generals problem is unsolvable
  • 55. Let’s use knowledge to prove it! But first… lots of formalism to get through L
  • 56. Road map for the proof: 1.  Semantics of modal logic 2.  Distributed system model 3.  A quick and easy lemma 4.  Big theorem: Common knowledge is not attainable via protocol 5.  Lemma 2: if the generals attack, they have common knowledge of the attack. 6.  Corollary: 2 generals is unsolvable
  • 58. Semantics: structures Formulae are well-formed, meaningless strings of symbols Structures give meaning to formulae (in the very narrow sense of making them all either true or false) S |= ϕ
  • 59. Semantics: propositional structures Propositional formula: S |= p ∧ q Need: 1.  a map S from variable names to T/F 2.  rules; e.g. S |= ϕ ∧ ψ iff S |= ϕ and S |= ψ
  • 60. Semantics: first-order structures First-order formula: S |= ∀x, dog(x) ⇒ big(x) ∧ likes(x, me) Need: 1.  S assigns “records” to dog, big and likes. 2.  Rules; e.g. S |= ∀xφ iff for all d ∈  |S|,  S[x  :=  d]  |=  φ  
  • 61. Semantics: first-order structures •  First-order logic: S |= ∀x, dog(x) ⇒ big(x) ∧ likes(x, me) dog   Rex   Fido   Rover   big   Rex   Fido   me   likes   Rex   me   Fido   me   Rover   me   me   me  
  • 62. couple good papers about using FO logic to program distributed systems
  • 63. Semantics – modal logic S |= (£¬p) ∧ (q ⇒ ¯r) Need: a structure that can interpret the propositional formulae under different modalities Kripke structure: (W, π, R) •  W is a set of worlds •  For each element of W, π is a propositional structure •  R is an accessibility relation among elements of W S1   S3  
  • 64. Semantics – modal logic Temporal logic S |= (£¬p) ∧ (q ⇒ ¯r)  q      r   r  q   S1   S3   S2   Kripke structure: (W, π, R)  
  • 65. Semantics – modal logic Epistemic logic S |= r ∧ ¬Kir ∧ Ki(Kjr or Kj¬r) ∧ Kjr ∧ ¬Kj¬Kir  q      r   r  q   S1   S3   S2   i   j   Kripke structure: (W, π, Ri)  
  • 66. a model of distributed systems (r,t) p1 p2 p3 p4 Idealized time }h(p4,r,t) A run r ∈ R
  • 67. Knowledge-based interpretations Knowledge interpretation: I = (R, π, {v1,v2,[..]}) Knowledge point: (I, r, t) R – a set of runs π – assigns a truth assignment to propositions for each point in R vi – A view function for R for some agent i (determined by h) Kripke structure: (W, π, R)  
  • 68. Truth in a knowledge interpretation (I,r,t) |= φ iff π(r,t)(φ) = true (If φ is a ground formula) (I,r,t) |= ¬φ iff (I,r,t) |= φ (I,r,t) |= φ ∧ ψ iff (I,r,t) |= φ and (I,r,t) |= ψ (I,r,t) |= Kiφ iff (I,r’,t’) |= φ for all (r’,t’) in R satisfying v(pi,r,t) = v(pi,r’,t’)   (I,r,t) |= Eφ iff (I,r’,t’) |= Kiφ for all pi (I,r,t) |= Cφ iff (I,r’,t’) |= Ekφ for all k
  • 69. choose-your-own-adventure •  If you’d like to gloss over the proof and skip to other applications of knowledge, turn to page 62 •  If you’d like to dive into the weeds, turn to page 54.
  • 70. Truth in a knowledge interpretation (I,r,t) |= Cφ iff (I,r’,t’) |= Ekφ for all k Fixed point axiom: Cφ = E(φ ∧ Cφ) Induction rule: From φ ⇒ E(φ ∧ ψ) infer φ ⇒ Cψ
  • 71. communication is not guaranteed   NG1: For all runs r and times t, there exists a run r’ extending (r,t) such that […] no messages are received in r’ at or after time t. NG2: If in run r processor pi does not receive any messages in the interval (t’,t), then there is a run r’ extending (r,t’) such that […] h(pi,r,t’’) = h(pi,r’,t’’) for all t’’ < t, and no processor pj != pi receives a message in r’ in the interval (t’,t).  
  • 72. Lemma 1 If, in two different runs (r and r’) of the same protocol, some h(p, r, t) = h(p, r’, t), then (I, r, t) |= Cφ iff (I, r’, t) |= Cφ Sorry, no proof today!
  • 73. Common knowledge is not attainable in a system in which communication is not guaranteed Take runs r and r- in R, with the same initial configuration, s.t. no messages are received in r- up till time t. Then (I,r,t) |= Cφ iff (I,r-,t) |= Cφ. Proof (by induction on d(r)*):   •  Base case: d(r)=0. h(p1,r,t) = h(p1,r-,t). By Lemma 1, (I,r,t) |= Cφ iff (I,r-,t) |= Cφ. *  d(r)  is  the  number  of  messages  received  in  run  r.  
  • 74. Common knowledge is not attainable in a system in which communication is not guaranteed Inductive case: d(r) = k+1. Let:   •  t’ < t -- the latest time a message is received in r before t. •  pj -- a processor that received a message at t’ •  pi –a processor (!= pj) By NG2, there is a run r’ extending (r,t’) s.t. h(pi,r,t’’)=h(pi,r’,t’’) for all t’’ <= t, and all processors (besides pi) receive no messages in the interval (t’, t). By construction, d(r’) <= k, so by the IH (I,r’,t) |= Cφ iff (I,r-,t) |= Cφ. But since h(pi,r,t) = h(pi,r’,t), by Lemma 1 (I,r’,t) |= Cφ iff (I,r,t) |= Cφ. So (I,r,t) |= Cφ iff (I,r-,t) |= Cφ. QED
  • 75. Common knowledge is not attainable in a system in which communication is not guaranteed Review: we showed that common knowledge cannot be gained (or lost) by exchanging messages. Corollary: the 2 generals will never attack. But we still need to prove one more lemma: Any correct protocol for coordinated attack has the property that whenever the generals attack, it is common knowledge that they are attacking.
  • 76. Lemma 2: coordinated attack requires common knowledge Let ψ = the generals are attacking Assume the generals (A and B) attack at (r*, t*) – we show that (I,r*,t*) |= Cψ. Pick an arbitrary point (r,t). We show ψ ⇒ Eψ is valid in R. •  If (I,r,t) |= ψ, then the generals attack at (r,t). Consider (r’,t’), in which A has the same history at (r,t). Since the protocol is deterministic (assumption), A must also attack in (r’,t’); since the protocol is correct, B does also, and so (I,r’,t’) |= ψ. It follows that (I,r,t) |= Eψ, so ψ ⇒ Eψ is valid in R. •  If (I,r,t) |= ¬ψ, then trivially ψ ⇒ Eψ is valid in R. By the induction rule, ψ ⇒ Cψ is valid in R
  • 77. Coup de grace ψ = the generals are attacking 1.  By assumption, Cψ does not hold if no messages are exchanged. 2.  By theorem 1, Cψ will never hold. 3.  By lemma 2, the generals cannot attack unless Cψ.  
  • 78. Phew. but…? Common knowledge is a prerequisite for agreement. Common knowledge is not attainable via protocol.
  • 79. Halpern: These results may seem paradoxical.
  • 80. Reality check Fragile assumptions on which the proofs rest: •  Deterministic protocol •  Simultaneous agreement is necessary •  “Communication not guaranteed” •  Lack of useful a priori common knowledge
  • 81. Bootstrapping common knowledge •  The ``weakest failure detector’’ •  Spanner’s global clock •  Sequence wraparound
  • 82. Applications of knowledge A correct distributed program achieves (nontrivial) distributed property X. Some tricky questions before we start coding: 1.  Is X even attainable? 2.  Cheapest protocol that gets me X? 3.  How should I implement it?
  • 83. lower bounds for protocols [Hadzilacos, PODS’87]: A knowledge-theoretic analysis of atomic commitment protocols 1.  All of the variants of 2pc ((de-)centralized, linear/nested, etc) are identical from a knowledge perspective 2.  All 2PC variants attain the minimum level of knowledge needed to commit 3.  3PC attains the minimum needed to commit without blocking 4.  Lower bound for messages: nested 2PC.
  • 84. A good paper about automatically choosing cheap coordination mechanisms
  • 85. Applications of knowledge A correct distributed program achieves (nontrivial) distributed property X. Some tricky questions before we start coding: 1.  Is X even attainable? 2.  Cheapest protocol that gets me X? 3.  How should I implement it?
  • 86. protocol implementation / synthesis •  Halpern and Fagin: knowledge-based programming [PODC’95]   case  of      K(Msg)  and  (KE(AckedMsg))  do  deliver(Msg)    K(Msg)  and  !KE(AckedMsg)  do  relay(Msg)       end   •  Matteo interlandi [Datalog2.0’11]: Knowlog: knowledge-enriched Dedalus  log(Tx_id,"abort")@next  :-­‐  Dvote(Vote,Tx_id),Vote=="no",                    par8cipants(X),transac8on(Tx_id,State),State=="vote-­‐req".    
  • 87. A good paper about Dedalus
  • 88. Monotonicity and knowledge Monotonic: the more you know, the more you know. Cϕ […] E3ϕ E2ϕ Eϕ Sϕ Dϕ ϕ  
  • 89. A good paper about monotonicity and distributed consistency
  • 90. Remember •  Knowledge is the dual of possibility •  Local knowledge dictates protocol behavior •  The purpose of protocols is obtaining a particular level of distributed knowledge •  Deep connections between semantic structures and system behavior •  Common knowledge is unattainable via protocol (but there is still hope)
  • 91. Protocols  climb  the  hierarchy   Cϕ […] E3ϕ E2ϕ Eϕ Sϕ Dϕ ϕ   Knowledge  Highway   E10ϕ          10   E100ϕ                    100   Cϕ  ∞  

Editor's Notes

  1. Protocols are so very often just mechanisms to ensure that a group has shared knowledge of a fact.
  2. State FLP. Today we’ll revisit these ideas. Find a common basis for a large family of impossibility results, lower bounds, and the raison d’etre of protocols: changing the state of distributed knowledge
  3. When I talk about DS in terms of what I know about what you know about what, the first thing you may think of is adversaries and byzantine systems. But forget about that for now. We’ll study non-byzantine protocols. Inconceivable? Hold on tight
  4. Due to the abundance of possible network behaviors and failures, it’s incredible hard to reason about program correctness. As we’ll see in a moment, KDP – the more we think is possible, the less we know. Knowing something is realizing that it’s impossible that it’s not so. Reasoning about Knowledge can give us unique insights into what’s fundamental about DS, protocols, etc.
  5. Now, this is only interesting if “knowledge” is a subtle thing…
  6. First example is trivial – knowledge is a NOOP. But it gets richer…
  7. Flat knowledge, global knowledge Intesection of knowledge across groups Global knowledge in ME
  8. But how deep does it go? It turns out it goes all the way.
  9. But how deep does it go? It turns out it goes all the way.
  10. But how deep does it go? It turns out it goes all the way.
  11. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  12. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  13. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  14. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  15. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  16. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  17. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  18. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  19. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  20. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  21. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  22. But how deep does it go? It turns out it goes all the way.
  23. But how deep does it go? It turns out it goes all the way.
  24. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  25. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  26. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  27. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  28. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  29. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  30. When could the sender clean up her buffer? When she knows that the receiver knows the message.
  31. But how deep does it go? It turns out it goes all the way.
  32. But how deep does it go? It turns out it goes all the way.
  33. But how deep does it go? It turns out it goes all the way.
  34. But how deep does it go? It turns out it goes all the way.
  35. Pause here. The first great and obvious application is proving impossibility results. We’ll spend a bunch of time here if you like!
  36. Wat is semantics?
  37. Wat is semantics?
  38. We “hold up” a formula to a structure too see if it’s truthy. The structures need (should) conform to our intuitions about real things. “semantics” are the rules that tell us (precisely) how to tell if a formula is true in a structure
  39. S needs to give us a domain (a universe of discourse) and relations over the domain (also, fussily, some constant & variable symbols, and sugaring in the form of functions. (this starts to seem like real life. The structure N of the natural numbers, along with the constant 0 and arithmetic functions is a nice structure) (S is essentially a database)
  40. S needs to give us a domain (a universe of discourse) and relations over the domain (also, fussily, some constant & variable symbols, and sugaring in the form of functions. (this starts to seem like real life. The structure N of the natural numbers, along with the constant 0 and arithmetic functions is a nice structure)
  41. “The system is deadlock-free, and every request eventually gets a response” This model does NOT satisfy the formula.
  42. “The system is deadlock-free, and every request eventually gets a response” This model does NOT satisfy the formula. The stroke of genius – to associate a kripke structure with a transition system – got emerson and clarke their 2007 turing award.
  43. The system is deadlock-free, and every request eventually gets a response This model does NOT satisfy the formula. Show a REAL kripke structure
  44. introduce (r,t). – a point in idealized time that cuts all the process lines. Sometimes we talk about R, the set of all runs for a DS, or for a protocol protocol in a model
  45. To make things simpler, let’s just say that v = h – ie, agents have infinite memory and are logically omniscient, in the sense that they “know” everything that follows from what they know.
  46. We have the structure – now we just need the precise rules. CHECK ME!!!!
  47. A processor knows something
  48. NG1: at some point, messages could stop being received forever. NG2: for any run in which some processor P receives no messages during some interval i-j there is another run in which no other processor does either. NG2 is broken.
  49. This is easy to show. Processor’s knowledge is just their initial state and their history. If even one processor can’t tell the difference btw two runs, then the formula holds – any state of group knowledge must be the same, since it involves her knowledge.